The format for the model output of some theories is not yet defined by the SMTLIB. The following document describes our proposal of such definitions. This format will be used by the model validator of the SMTLIB for the experimental model validation track of the SMTCOMP 2023.
Some theory functions are only partially defined, e.g., division by zero. The SMT semantics states that a benchmark is sat if there is an extension of these partial functions to a total function, i.e., the value of the function can be chosen by the solver. This begs the questions how a solver should describe the model it chose.
We propose that solvers should give the values similar as for
uninterpreted functions using
define-fun. The given function definition
must coincide on the defined inputs with the values given by the
theory. To achieve this, the function definition may call the original theory
function using the same name as the function that is defined.
In the model validation track, solvers MUST give the values for
an undefined input, if it affects the satisfiability of the benchmark.
A solver SHOULD give a concrete value for all undefined inputs. A
simple way to achieve this is with an
ite expression that checks if
the input leads to undefinedness and provides a concrete expression in
that case and calls the original theory function in the other case.
As an example, consider the theories of non-linear arithmetic and data types and the following definition of an algebraic data type for lists in the input problem:
(declare-datatypes ((list 1)) ((par (alpha) ( (nil) (cons (car alpha) (cdr (list alpha))) ))))
The partially defined theory functions may be defined using the following definitions in the model:
(define-fun div ((a Int) (b Int)) Int (ite (= b 0) (ite (= a 0) 5 0) (div a b))) (define-fun car ((a (List Int))) (List Int) (match a ((cons hd tl) hd) (nil 42))) (define-fun cdr ((a (List Int))) Int (ite ((_ is cons) a) (cdr a) a))
cdr should not be considered to be recursive function.
Instead the definition shadows the (partially defined) theory function and
the definition calls the original theory function in the function body.
The model checker
is lenient, it will complain only if it needs a value in the uninterpreted
domain of a partial function. For example for
(assert (= 0.5 (div 1.0 x))),
the definition of
div can be omitted if the solver chose
2.0 as the
x in the model, but it can’t be omitted if it chose
Representations of (algebraic
numbers)[https://en.wikipedia.org/wiki/Algebraic_number] are needed for
representing models of
QF_NRA. Their representation usually consists of two
parts, one polynomial and a way to specify which root of the polynomial is the
represented algebraic numbers.
For the polynomial there are two main choices:
For specifying the roots:
The ordering can be smallest to greatest however it doesn’t extend to complex values. Moreover, multiplicities must be taken into account. However, it offers the possibility to have a unique representation for an algebraic number.
We identified two opposing objectives:
The first objective allows checking syntactically which values are equal in a model. The human readability means that a human can get an understanding of the value of the number just by reading the value. We propose two ways to write values of algebraic numbers that corresponding to one objective each
root-of-with-interval. In both case the polynomial with integer coefficient is represented by the list of coefficients in ascending order, i.e., from the coefficient of the constant term to the leading coefficient. For example, $x^2 - 2$ is represented as
((- 2) 0 1). For uniqueness, the coefficients should be coprime integers and the leading coefficient should be positive.
(root-of-with-ordering (coeffs p_0 p_1 ... p_n) i)represents the
i-th root ordered with multiplicity from the smallest to greatest of the polynomial
(p_0 p_1 ... p_n). Here,
iis a numeral (non-negative integer) and is
0for the smallest root of the polynomial. The polynomial must be the unique reduced minimal polynomial, in particular, it must only have simple roots.
(root-of-with-interval (coeffs p_0 p_1 ... p_n) min max)represents the unique root between
maxof the polynomial
(p_0 p_1 ... p_n).
iis an integer.
maxare rational model values, e.g.
(0.0 (/ 1.0 2.0))for the interval
((- 1.0) (/ (- 1.0) 2.0))for the interval [-1, -.5].
The constraint on the uniqueness of the polynomial in
root-of-with-ordering could perhaps be removed.
coeffs, but it is necessary in order to be an SMTLIB term.
(min max)instead of
min max, but it is necessary in order to be an SMTLIB term.
Models for arrays should be represented using the
(define-fun b () (Array Int Real) (store (store ((as const (Array Int Real)) 2.0) 0 1.0) 1 3.0))
There were two alternative suggestions but we rejected them for various reasons at least for this years conference. We will still outline them here to explain our reasoning. The alternatives are:
(define-fun b () (Array Int Real) (_ as-array k!1)) (define-fun k!1 ((x!0 Int)) Real (ite (= x!0 1) 3.0 (ite (= x!0 0) 1.0 2.0)))
(define-fun b () (Array Int Real) (store (store (as @array0 (Array Int Real)) 0 1.0) 1 3.0))
Note that here the value is not uniquely defined. For our purpose,
the array values must explicitly use store to define values for all
indices that are read by any
select in the benchmark.
Different model values for arrays are considered
to be not equal like in the UF logic.
The const/store representation gives directly a constant term (that use the symbol
const that is not defined by the SMTLIB format) when the first alternative requires to define an additional function.
The first alternative is more general
and can handle models for problems with quantifiers. Since SMT-LIB 3 should
introduce anonymous functions we could backport the feature for the
definition of model of arrays with the addition of the function
as-array to convert from a function
(-> A B) to an array
(Array A B). We would have
(define-fun b () (Array Int Real) (as-array (lambda ((x!0 Int)) (ite (= x!0 1) 3.0 (ite (= x!0 0) 1.0 2.0)))))
However, using generic functions for defining arrays makes it harder for
the model validator to evaluate array equality (extensionality).
So in order to keep the validator simple for quantifier free formulas,
we require the first way of defining array models in quantifier free logic.
const function is only visible in the array models and not part of the
theory. A benchmark problem must not contain the
const function in an
To sum up, in quantifier free logic, models for arrays should be represented using